Imported .svg will not extrude
Use the Extrude option for curves. This let's you keep the SVG as adjustable vector art until you need to convert it to a mesh.


Import your SVG to Blender. Imported vector art normally comes in as a Blender curve. Depending on the complexity of your SVG, you may need to select all curves with the file and use cntl-J to combine them into a complex curve. Scale it to suit.
Select the SVG, and go to the Curve panel. The Curve panel offers a range of different options you can use to adjust the resolution, fill type or... lots of things.
Adjust Geometry > Extrude. This will extrude your SVG at 90-degrees to the plane of the SVG.
(Optional) Shape > Fill > Both. This might already be on Fill, but this will 'cap' closed curves in the SVG, creating solid pillars of your curve.
(Totally optional) Shape > Resolution > Preview U > [some number]. Curves have infinite resolution, but eventually you will probably need to convert the curve to a mesh. The 'U' number is the number of segments each line segment will be divided. Use this carefully if you have a combination of very long and very short segments. You may wish to subdivide long segments (or combine short ones) before you apply the 'U' value.
Have fun. Tip: You can adjust the extrude shape using another curve, providing an interesting way to shape your extrusion.


Demo video: Extruding SVGs